Integrand size = 28, antiderivative size = 240 \[ \int \frac {(e x)^{3/2} \left (a+b x^2\right )^2}{\sqrt {c+d x^2}} \, dx=\frac {2 \left (77 a^2 d^2+5 b c (9 b c-22 a d)\right ) e \sqrt {e x} \sqrt {c+d x^2}}{231 d^3}-\frac {2 b (9 b c-22 a d) (e x)^{5/2} \sqrt {c+d x^2}}{77 d^2 e}+\frac {2 b^2 (e x)^{9/2} \sqrt {c+d x^2}}{11 d e^3}-\frac {c^{3/4} \left (77 a^2 d^2+5 b c (9 b c-22 a d)\right ) e^{3/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{231 d^{13/4} \sqrt {c+d x^2}} \]
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Time = 0.16 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {475, 470, 327, 335, 226} \[ \int \frac {(e x)^{3/2} \left (a+b x^2\right )^2}{\sqrt {c+d x^2}} \, dx=-\frac {c^{3/4} e^{3/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (77 a^2 d^2+5 b c (9 b c-22 a d)\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{231 d^{13/4} \sqrt {c+d x^2}}+\frac {2 e \sqrt {e x} \sqrt {c+d x^2} \left (77 a^2 d^2+5 b c (9 b c-22 a d)\right )}{231 d^3}-\frac {2 b (e x)^{5/2} \sqrt {c+d x^2} (9 b c-22 a d)}{77 d^2 e}+\frac {2 b^2 (e x)^{9/2} \sqrt {c+d x^2}}{11 d e^3} \]
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Rule 226
Rule 327
Rule 335
Rule 470
Rule 475
Rubi steps \begin{align*} \text {integral}& = \frac {2 b^2 (e x)^{9/2} \sqrt {c+d x^2}}{11 d e^3}+\frac {2 \int \frac {(e x)^{3/2} \left (\frac {11 a^2 d}{2}-\frac {1}{2} b (9 b c-22 a d) x^2\right )}{\sqrt {c+d x^2}} \, dx}{11 d} \\ & = -\frac {2 b (9 b c-22 a d) (e x)^{5/2} \sqrt {c+d x^2}}{77 d^2 e}+\frac {2 b^2 (e x)^{9/2} \sqrt {c+d x^2}}{11 d e^3}-\frac {1}{77} \left (-77 a^2-\frac {5 b c (9 b c-22 a d)}{d^2}\right ) \int \frac {(e x)^{3/2}}{\sqrt {c+d x^2}} \, dx \\ & = \frac {2 \left (77 a^2+\frac {5 b c (9 b c-22 a d)}{d^2}\right ) e \sqrt {e x} \sqrt {c+d x^2}}{231 d}-\frac {2 b (9 b c-22 a d) (e x)^{5/2} \sqrt {c+d x^2}}{77 d^2 e}+\frac {2 b^2 (e x)^{9/2} \sqrt {c+d x^2}}{11 d e^3}-\frac {\left (c \left (77 a^2+\frac {5 b c (9 b c-22 a d)}{d^2}\right ) e^2\right ) \int \frac {1}{\sqrt {e x} \sqrt {c+d x^2}} \, dx}{231 d} \\ & = \frac {2 \left (77 a^2+\frac {5 b c (9 b c-22 a d)}{d^2}\right ) e \sqrt {e x} \sqrt {c+d x^2}}{231 d}-\frac {2 b (9 b c-22 a d) (e x)^{5/2} \sqrt {c+d x^2}}{77 d^2 e}+\frac {2 b^2 (e x)^{9/2} \sqrt {c+d x^2}}{11 d e^3}-\frac {\left (2 c \left (77 a^2+\frac {5 b c (9 b c-22 a d)}{d^2}\right ) e\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{231 d} \\ & = \frac {2 \left (77 a^2+\frac {5 b c (9 b c-22 a d)}{d^2}\right ) e \sqrt {e x} \sqrt {c+d x^2}}{231 d}-\frac {2 b (9 b c-22 a d) (e x)^{5/2} \sqrt {c+d x^2}}{77 d^2 e}+\frac {2 b^2 (e x)^{9/2} \sqrt {c+d x^2}}{11 d e^3}-\frac {c^{3/4} \left (77 a^2+\frac {5 b c (9 b c-22 a d)}{d^2}\right ) e^{3/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{231 d^{5/4} \sqrt {c+d x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 11.17 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.79 \[ \int \frac {(e x)^{3/2} \left (a+b x^2\right )^2}{\sqrt {c+d x^2}} \, dx=\frac {(e x)^{3/2} \left (\frac {2 \sqrt {x} \left (c+d x^2\right ) \left (77 a^2 d^2+22 a b d \left (-5 c+3 d x^2\right )+3 b^2 \left (15 c^2-9 c d x^2+7 d^2 x^4\right )\right )}{d^3}-\frac {2 i c \left (45 b^2 c^2-110 a b c d+77 a^2 d^2\right ) \sqrt {1+\frac {c}{d x^2}} x \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {i \sqrt {c}}{\sqrt {d}}}}{\sqrt {x}}\right ),-1\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {d}}} d^3}\right )}{231 x^{3/2} \sqrt {c+d x^2}} \]
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Time = 3.12 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.01
method | result | size |
risch | \(\frac {2 \left (21 b^{2} d^{2} x^{4}+66 x^{2} a b \,d^{2}-27 x^{2} b^{2} c d +77 a^{2} d^{2}-110 a b c d +45 b^{2} c^{2}\right ) x \sqrt {d \,x^{2}+c}\, e^{2}}{231 d^{3} \sqrt {e x}}-\frac {c \left (77 a^{2} d^{2}-110 a b c d +45 b^{2} c^{2}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) e^{2} \sqrt {e x \left (d \,x^{2}+c \right )}}{231 d^{4} \sqrt {d e \,x^{3}+c e x}\, \sqrt {e x}\, \sqrt {d \,x^{2}+c}}\) | \(242\) |
elliptic | \(\frac {\sqrt {e x \left (d \,x^{2}+c \right )}\, \sqrt {e x}\, \left (\frac {2 b^{2} e \,x^{4} \sqrt {d e \,x^{3}+c e x}}{11 d}+\frac {2 \left (2 a b \,e^{2}-\frac {9 b^{2} e^{2} c}{11 d}\right ) x^{2} \sqrt {d e \,x^{3}+c e x}}{7 d e}+\frac {2 \left (a^{2} e^{2}-\frac {5 \left (2 a b \,e^{2}-\frac {9 b^{2} e^{2} c}{11 d}\right ) c}{7 d}\right ) \sqrt {d e \,x^{3}+c e x}}{3 d e}-\frac {\left (a^{2} e^{2}-\frac {5 \left (2 a b \,e^{2}-\frac {9 b^{2} e^{2} c}{11 d}\right ) c}{7 d}\right ) c \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{3 d^{2} \sqrt {d e \,x^{3}+c e x}}\right )}{e x \sqrt {d \,x^{2}+c}}\) | \(304\) |
default | \(-\frac {e \sqrt {e x}\, \left (-42 b^{2} d^{4} x^{7}+77 \sqrt {-c d}\, \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c \,d^{2}-110 \sqrt {-c d}\, \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{2} d +45 \sqrt {-c d}\, \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{3}-132 a b \,d^{4} x^{5}+12 b^{2} c \,d^{3} x^{5}-154 a^{2} d^{4} x^{3}+88 x^{3} d^{3} b a c -36 b^{2} c^{2} d^{2} x^{3}-154 a^{2} c \,d^{3} x +220 a b \,c^{2} d^{2} x -90 b^{2} d x \,c^{3}\right )}{231 x \sqrt {d \,x^{2}+c}\, d^{4}}\) | \(405\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.53 \[ \int \frac {(e x)^{3/2} \left (a+b x^2\right )^2}{\sqrt {c+d x^2}} \, dx=-\frac {2 \, {\left ({\left (45 \, b^{2} c^{3} - 110 \, a b c^{2} d + 77 \, a^{2} c d^{2}\right )} \sqrt {d e} e {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right ) - {\left (21 \, b^{2} d^{3} e x^{4} - 3 \, {\left (9 \, b^{2} c d^{2} - 22 \, a b d^{3}\right )} e x^{2} + {\left (45 \, b^{2} c^{2} d - 110 \, a b c d^{2} + 77 \, a^{2} d^{3}\right )} e\right )} \sqrt {d x^{2} + c} \sqrt {e x}\right )}}{231 \, d^{4}} \]
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Result contains complex when optimal does not.
Time = 9.24 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.60 \[ \int \frac {(e x)^{3/2} \left (a+b x^2\right )^2}{\sqrt {c+d x^2}} \, dx=\frac {a^{2} e^{\frac {3}{2}} x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 \sqrt {c} \Gamma \left (\frac {9}{4}\right )} + \frac {a b e^{\frac {3}{2}} x^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{\sqrt {c} \Gamma \left (\frac {13}{4}\right )} + \frac {b^{2} e^{\frac {3}{2}} x^{\frac {13}{2}} \Gamma \left (\frac {13}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {13}{4} \\ \frac {17}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 \sqrt {c} \Gamma \left (\frac {17}{4}\right )} \]
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\[ \int \frac {(e x)^{3/2} \left (a+b x^2\right )^2}{\sqrt {c+d x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2} \left (e x\right )^{\frac {3}{2}}}{\sqrt {d x^{2} + c}} \,d x } \]
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\[ \int \frac {(e x)^{3/2} \left (a+b x^2\right )^2}{\sqrt {c+d x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2} \left (e x\right )^{\frac {3}{2}}}{\sqrt {d x^{2} + c}} \,d x } \]
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Timed out. \[ \int \frac {(e x)^{3/2} \left (a+b x^2\right )^2}{\sqrt {c+d x^2}} \, dx=\int \frac {{\left (e\,x\right )}^{3/2}\,{\left (b\,x^2+a\right )}^2}{\sqrt {d\,x^2+c}} \,d x \]
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